Controllers
Controllers
Operational amplifiers, which is often shorten to op amps, are extensively used to amplify
signals in electronic circuits. The standard op amp looks like the following;
It is common practice to have a ground that is 0 volts, and ev

Type of Change
1) Blocks in series
2) Feedback Loop
3) Feed forward
Loop
4) Removing a
block from a
feedback loop
5) Removing a
block from a feed
forward loop
6) Moving a
summation point
to in front of a
block
7) Moving a
summation point
to beyond a block

Slope and Deflection of Beams
The Deflection Curve
An effect of loading an initially straight beam is to deform it into a
curved shape. The deflection can be expressed in terms of the
deflection of the beam from its initially unloaded position. The
conven

Block Diagram Models
The arrows show the direction of signal flow. Usually lowercase for function of time i.e.
i(t) and in the s domain, they are represented by capital letters followed by s, i.e. I(s). The
summing point is where signals are algebraically

Magnitude(dB)
Phase(deg)
180
1
10
90
0
90
60
180
40
20
0
20
40
60
0
10
Frequency(rad/s)
1
10
BodePlot
2
10
3
10
Bode diagram for system
T .F .
25 s 4
is as follows;
s s 10
The transfer function above can be simplified to the standard form that we requi

Pole and Zeros
There are a number of different ways to represent a transfer function. The classic form is as a
polynomial which is
numerator
6 4 4 4 44
7 4 4 4 4 48
m
m 1
b s bm1s L b1s1 b0
G s m p
a p s a p 1s p 1 L a1s1 a0
1 4 4 4 44 2 4 4 4 4 43
denomi

SYSTEM MODELS
Introduction
In order to analyze control systems, mathematical models are used to represent the
elements used in such systems. These are the equations representing the input and output
and the relationship between them. The basis for any mat

Laplace Transforms
The Laplace transform is a method where differential equations which involve time are transformed
into simple algebraic relationships, which do not involve time, where we can carry out our standard
algebraic manipulations of the quantit

Gears
If we look at the diagram below, we see two gears that are intermeshed. The input gear has
radius r1 and N1 teeth is rotated through an angle 1 (t ) due to an torque T1 (t ). The output gear
has radius r2 and N 2 teeth is rotated through an angle 2

Mechanics and Materials3
Paddy Mac Namara
MECH3001
Rev: B
Cylindrical shells
Mechanical Science, W.Bolton. ISBN 0 632 03579 X
Thin cylindrical shell
When a closed cylindrical shell is subject to an internal gauge pressure, P
perhaps due to it containing a